Projectile Max Height Calculator

Projectile motion: h_max = (v₀·sin θ)² / (2g)

In an oblique launch with no air resistance, the path traced out is a parabola, and three formulas describe it. Maximum height is h_max = (v₀·sin θ)² / (2g), range is R = v₀²·sin(2θ) / g, and the time to rise to the apex is t_rise = v₀·sin θ / g; the whole flight lasts twice that. On a horizontal plane the range peaks at θ = 45°, because that makes sin(2·45°) = 1. Try v₀ = 20 m/s and θ = 30°: you get h_max ≈ 5.1 m, R ≈ 35.3 m, and a total time near 2 s. Now look at real-world projectiles such as bullets, cannonballs, or baseballs. Air drag rewrites the rules. The best angle for distance drops to roughly 35°, and the arc loses its symmetry, with a descent steeper than the climb that produced it.

Applications

Ballistics turns to it for forensic analysis and defense work. Sports do too, from free kicks to the arc of a basketball shot to the javelin throw, alongside rocket and missile engineering, physics-based games like Angry Birds and Worms, and the biomechanics taught in physical education.

FAQ

Why does 45° maximize range? The range formula carries a sin(2θ) term, and that term hits its largest value when 2θ = 90°, meaning θ = 45°. Step above or below that angle and the range falls off symmetrically.

Does the projectile's mass affect its trajectory? Not in a vacuum, where mass cancels out and gravity accelerates everything alike. Add air resistance and it does matter: heavier projectiles feel the drag less.

Why is the optimal angle smaller for real projectiles? Drag scales with the square of velocity, so a steeper launch keeps the projectile in flight longer while it bleeds speed, and that eats into the range.